The Stacks project

Lemma 105.11.4. Let $f : \mathcal{X} \to \mathcal{Y}$ and $h : \mathcal{U} \to \mathcal{X}$ be morphisms of algebraic stacks. Assume that $\mathcal{Y}$ is locally Noetherian, that $f$ is locally of finite type and quasi-separated, that $h$ is of finite type, and that the image of $|h| : |\mathcal{U}| \to |\mathcal{X}|$ is dense in $|\mathcal{X}|$. If given any $2$-commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-u \ar[d]_ j & \mathcal{U} \ar[r]_ h & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[rr]^-y \ar@{..>}[rru] & & \mathcal{Y} } \]

where $A$ is a discrete valuation ring with field of fractions $K$ and $\gamma : y \circ j \to f \circ h \circ u$, the category of dotted arrows is either empty or a setoid with exactly one isomorphism class, then $f$ is separated.

Proof. We have to prove $\Delta $ is a proper morphism. Assume first that $\Delta $ is separated. Then we may apply Lemma 105.11.3 to the morphisms $\mathcal{U} \to \mathcal{X}$ and $\Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$. Observe that $\Delta $ is quasi-compact as $f$ is quasi-separated. Of course $\Delta $ is locally of finite type (true for any diagonal morphism, see Morphisms of Stacks, Lemma 100.3.3). Finally, suppose given a $2$-commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-u \ar[d]_ j & \mathcal{U} \ar[r]_ h & \mathcal{X} \ar[d]^\Delta \\ \mathop{\mathrm{Spec}}(A) \ar[rr]^-y \ar@{..>}[rru] & & \mathcal{X} \times _\mathcal {Y} \mathcal{X} } \]

where $A$ is a discrete valuation ring with field of fractions $K$ and $\gamma : y \circ j \to \Delta \circ h \circ u$. By Morphisms of Stacks, Lemma 100.41.1 and the assumption in the lemma we find there exists a unique dotted arrow. This proves the last assumption of Lemma 105.11.3 holds and the result follows.

In the general case, it suffices to prove $\Delta $ is separated since then we'll be back in the previous case. In fact, we claim that the assumptions of the lemma hold for

\[ \mathcal{U} \to \mathcal{X} \quad \text{and}\quad \Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X} \]

Namely, since $\Delta $ is representable by algebraic spaces, the category of dotted arrows for a diagram as in the previous paragraph is a setoid (see for example Morphisms of Stacks, Lemma 100.39.2). The argument in the preceding paragraph shows these categories are either empty or have one isomorphism class. Thus $\Delta $ is separated. $\square$


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